principal infinity-bundle in nLab

Context

Bundles

Cohomology

• Idea
• Definition
• GG-principal ∞\infty-bundles
• Connections on GG-principal ∞\infty-bundles
• Concrete realizations
• In topological spaces
• In simplicial sets / Kan complexes
• In a petit (∞,1)(\infty,1)-topos
• In a gros (∞,1)(\infty,1)-topos
• Smooth principal ∞\infty-bundles
• Examples
• Ordinary principal bundles
• Circle nn-bundles
• Bundle gerbes
• Normal morphisms of ∞\infty-groups
• Related concepts
• References

Definition

(principal GG-action)

Let GG be a group object in the (∞,1)-topos H\mathbf{H}. A principal action of GG on a morphism (P→X)∈H(P \to X) \in \mathbf{H} is a groupoid object P//GP//G that sits over *//G*//G in that we have a morphism of simplicial diagrams Δ op→H\Delta^{op} \to \mathbf{H}

⋮ ⋮ P×G×G →(p 2,p 3) G×G ↓↓↓ ↓↓↓ P×G →p 2 G ↓↓ ↓↓ P → * \array{ \vdots && \vdots \\ P \times G \times G &\stackrel{(p_2, p_3)}{\to}& G \times G \\ \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow \\ P \times G &\stackrel{p_2}{\to}& G \\ \downarrow\downarrow && \downarrow\downarrow \\ P &\stackrel{}{\to}& {*} }

in H\mathbf{H};

and such that P→XP \to X exhibits the (∞,1)-colimit

X≃lim →(P//G:Δ op→H) X \simeq \lim_\to (P//G : \Delta^{op} \to \mathbf{H})

called the base space over which the action takes place.

Proposition

The GG-principal action as defined above satisfies the principality condition in that we have an equivalence of groupoid objects

⋮ ⋮ P× XP× XP →≃ P×G×G ↓↓↓ ↓↓↓ P× XP →≃ P×G ↓↓ ↓↓ P →≃ P. \array{ \vdots && \vdots \\ P \times_X P \times_X P &\stackrel{\simeq}{\to}& P \times G \times G \\ \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow \\ P \times_X P &\stackrel{\simeq}{\to}& P \times G \\ \downarrow\downarrow && \downarrow\downarrow \\ P &\stackrel{\simeq}{\to}& P } \,.

Proposition

For X→BGX \to \mathbf{B}G any morphism, its homotopy fiber P→XP \to X is canonically equipped with a principal GG-action with base space XX.

Proof

First we show that we have a morphism of simplicial diagrams

⋮ ⋮ ⋮ P× XP× XP →≃ P×G×G → G×G ↓↓↓ ↓↓↓ ↓↓↓ P× XP →≃ P×G →p 2 G ↓↓ ↓↓ ↓↓ P →= P → * ↓ ↓ ↓ X →= X →g BG, \array{ \vdots && \vdots && \vdots \\ P \times_X P \times_X P &\stackrel{\simeq}{\to}& P \times G \times G &\to& G \times G \\ \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow \\ P \times_X P &\stackrel{\simeq}{\to}& P \times G &\stackrel{p_2}{\to}& G \\ \downarrow\downarrow && \downarrow\downarrow && \downarrow\downarrow \\ P &\stackrel{=}{\to}& P &\stackrel{}{\to}& {*} \\ \downarrow && \downarrow && \downarrow \\ X &\stackrel{=}{\to}& X &\stackrel{g}{\to}& \mathbf{B}G } \,,

with the right square swhere the left horizontal morphisms are equivalences, as indicated. We proceed by induction through the height of this diagram.

The defining (∞,1)-pullback square for P× XPP \times_X P is

P× XP → P ↓ ↓ P → X \array{ P \times_X P &\to& P \\ \downarrow && \downarrow \\ P &\to& X }

To compute this, we may attach the defining (∞,1)(\infty,1)-pullback square of PP to obtain the pasting diagram

P× XP → P → * ↓ ↓ ↓ P → X → BG. \array{ P \times_X P &\to& P &\to& {*} \\ \downarrow && \downarrow && \downarrow \\ P &\to& X &\to& \mathbf{B}G \,. }

and use the pasting law for pullbacks, to conclude that P× XPP \times_X P is the pullback

P× XP → → * ↓ ↓ P → X → BG. \array{ P \times_X P &\to& &\to& {*} \\ \downarrow && && \downarrow \\ P &\to& X &\to& \mathbf{B}G \,. }

By definition of PP as the homotopy fiber of X→BGX \to \mathbf{B}G, the lower horizontal morphism is equivbalent to P→*→BGP \to {*} \to \mathbf{B}G, so that P× XPP \times_X P is also equivalent to the pullback

P× XP → → * ↓ ↓ P → * → BG. \array{ P \times_X P &\to& &\to& {*} \\ \downarrow && && \downarrow \\ P &\to& {*} &\to& \mathbf{B}G \,. }

This finally may be computed as the pasting of two pullbacks

P× XP ≃ P×G → G → * ↓ ↓ ↓ P → * → BG. \array{ P \times_X P &\simeq& P \times G &\to& G &\to& {*} \\ &&\downarrow && \downarrow && \downarrow \\ &&P &\to& {*} &\to& \mathbf{B}G \,. }

of which the one on the right is the defining one for GG and the remaining one on the left is just an (∞,1)-product.

Proceeding by induction from this case we find analogously that P × X n+1≃P×G × nP^{\times_X^{n+1}} \simeq P \times G^{\times_n}: suppose this has been shown for (n−1)(n-1), then the defining pullback square for P × X n+1P^{\times_X^{n+1}} is

P× XP × X n → P ↓ ↓ P × X n → X. \array{ P \times_X P^{\times_X^n} &\to& P \\ \downarrow && \downarrow \\ P^{\times_X^n}&\to& X } \,.

We may again paste this to obtain

P× XP × X n → P → * ↓ ↓ ↓ P × X n → X → BG \array{ P \times_X P^{\times_X^n} &\to& P &\to& * \\ \downarrow && \downarrow && \downarrow \\ P^{\times_X^n}&\to& X &\to& \mathbf{B}G }

and use from the previous induction step that

(P × X n→X→BG)≃(P × X n→*→BG) (P^{\times_X^n} \to X \to \mathbf{B}G) \simeq (P^{\times_X^n} \to * \to \mathbf{B}G)

to conclude the induction step with the same arguments as before.

This shows that P//GP//G is the Cech nerve of P→XP \to X. It remains to show that indeed X=lim → nP×G × nX = {\lim_\to}_n P \times G^{\times^n}. For this notice that *→BG* \to \mathbf{B}G is an effective epimorphism in an (infinity,1)-category. Hence so is P→XP \to X. This proves the claim, by definition of effective epimorphism.

using this we have

X ≃BG∏ BGX ≃(lim → nG × n)∏ BGX ≃lim → n(G × n∏ BGX) ≃lim → n(P×G × n) ≃lim →P//G. \begin{aligned} X & \simeq \mathbf{B}G \prod_{\mathbf{B}G} X \\ & \simeq \left({\lim_{\to}}_n G^{\times^n}\right) \prod_{\mathbf{B}G} X \\ & \simeq {\lim_{\to}}_n ( G^{\times^n} \prod_{\mathbf{B}G} X ) \\ & \simeq {\lim_\to}_n ( P\times G^{\times^n} ) \\ & \simeq {\lim_\to} P//G \end{aligned} \,.

Definition

We say a GG-principal action of GG on PP over XX is a GG-principal ∞-bundle if the colimit over P//G→*//GP//G \to *//G produces a pullback square: the bottom square in

⋮ ⋮ P×G×G → G×G ↓↓↓ ↓↓↓ P×G →p 2 G ↓↓ ↓↓ P → * ↓ ↓ X=lim →(P×G •) →g BG=lim →(G •). \array{ \vdots && \vdots \\ P \times G \times G &\to& G \times G \\ \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow \\ P \times G &\stackrel{p_2}{\to}& G \\ \downarrow\downarrow && \downarrow\downarrow \\ P &\stackrel{}{\to}& {*} \\ \downarrow && \downarrow \\ X = \lim_\to (P \times G^\bullet) &\stackrel{g}{\to}& \mathbf{B}G = \lim_\to( G^\bullet) } \,.

Proof

The arrow category H I\mathbf{H}^I is still an (infinity,1)-topos and hence the Giraud-Lurie axioms still hold. This means that by the discussion at groupoid object in an (infinity,1)-category (using the statement below HTT, cor. 6.2.3.5) we have an equivalence

Grpd(H I)≃(H I) eff I Grpd(\mathbf{H}^I) \simeq (\mathbf{H}^{I})^{I}_{eff}

between groupoid objects in H I\mathbf{H}^I and effective epimorphisms in the arrow category.

Notice that groupoid objects and effective epis in H I\mathbf{H}^I are given objectwise over the two objects of the interval I=Δ[1]I = \Delta[1].

Restricting this equivalence along the inclusion

H(X,BG)↪(H I) I \mathbf{H}(X, \mathbf{B}G) \hookrightarrow (\mathbf{H}^I)^I

given by sending a cocycle to its homotopy fiber diagram

(X→BG)↦(P → * ↓ ↓ X → BG) (X \to \mathbf{B}G) \mapsto \left( \array{ P &\to& * \\ \downarrow && \downarrow \\ X &\to& \mathbf{B}G } \right)

therefore yields precisely the equivalence in question

GBund(X) ↪ Grpd(H I) ↓ ≃ ↓ ≃ H(X,BG) ↪hofib (H I) I. \array{ G Bund(X) &\hookrightarrow& Grpd(\mathbf{H}^I) \\ \downarrow^\simeq && \downarrow^\simeq \\ \mathbf{H}(X, \mathbf{B}G) &\stackrel{hofib}{\hookrightarrow}& (\mathbf{H}^I)^I } \,.

Theorem

Given the above, the diagram

P → * ↓ ↓ X →g BG \array{ P &\to& {*} \\ \downarrow && \downarrow \\ X &\stackrel{g}{\to}& \mathbf{B}G }

is a homotopy pullback (i.e. defines a fibration sequence).

Proof

This is originally due to

• D. Quillen, Higher algebraic K-theory I, Springer Lecture notes in Math. 341 (1973) 85–147.

The statement is reproduced in section IV of

• P. G. Goerss and J. F. Jardine, 1999, Simplicial Homotopy Theory, number 174 in Progress in Mathematics, Birkhauser. (ps)

Theorem

Let now CC be a category and for

ρ P:C→SSet \rho_P : C \to SSet

a functor that sends all morphisms to weak equivalences of simplicial sets.

Let now for each object c∈Cc \in C

P c:=ρ C(c) P_c := \rho_C(c)

be the “bundle of cc-fibers”.

Then for each cc the diagram

P c → * ↓ ↓ *↦c X →g C \array{ P_c &\to& {*} \\ \downarrow && \downarrow^{{*} \mapsto c} \\ X &\stackrel{g}{\to}& C }

is a homotopy pullback (i.e. defines a fibration sequence).